key21fa02 - COT3100.01, Fall 2002 S. Lang (10/31/2002)...

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COT3100.01, Fall 2002 S. Lang (10/31/2002) Answer key to Test #2 Nov. 4, 2002 Part I. (39 pts., 3 pts. each) True/False, or short-answer, questions. (No explanation needed and no partial credit given.) 1. Let N denote the set of natural numbers {0, 1, 2, …}, and let A N and B N . Suppose set A contains a smallest value and set B contains a smallest value. Thus, the set A B contains a smallest value. ( False , since the set A B may be empty) 2. Suppose A and B are both finite sets. If A B and | A | = | B |, then A = B . ( True , since A B , B = A ( B A ) is true, which implies | B | = | A | + | B A | because A ( B A ) = ; thus, | B A | = 0 because | A | = | B | by assumption. Therefore, the set B A = , so A = B is true since A B .) 3. Suppose n 1 is an integer. Then 2 n > C ( n , k ) for any integer k with n k 1. ( True , by the formula 2 n = C ( n , 0) + C ( n , 1) + …+ C ( n , n ) > C ( n , k ) for any k , n k 1.) 4. If C ( n , 5) = C ( n , 3), then the exact numerical value of n = 8, because C (8, 5) = C (8, 8 – 5) = C (8, 3). 5. Suppose n 1 is an integer, and 1 + 2 + … + n = 20100. The exact numerical value of n = 200, because by the formula 1 + 2 + … + n = 2 ) 1 ( + n n = 20100, n ( n + 1) = 20100 2 = 200 201. 6. In how many ways can 10 movie tickets (to the same movie) be divided between 3 persons where each get at least one ticket? C (9, 2), by lining up the 10 tickets in a row which shows 9 gaps, and choosing 2 out of 9 divides the 10 tickets into 3 groups corresponding to ways of dividing the tickets to 3 persons. (This is the same as the example of dividing 10 apples among 3 persons, as in the course notes.) 7. In how many ways can the 9 letters in the word “PAPERTAPE” (and only these letters) be re-arranged? ! 2 ! 2 ! 3 ! 9 , because there are 9 letters for the permutations, consisting of 3 Ps, 2 As, 2 Es, and 1 of each remaining letters R and T. 8. A pocket contains 5 coins: 2 quarters, 2 dimes, and 1 penny. When blindly picking two coins out of the pocket, is it more likely to get 1 quarter and 1 dime, or more likely to get 1 penny and 1 dime? Getting 1 quarter and 1 dime is more likely because there are 2 2 = 4 many ways to select 1 quarter and 1 dime, compared to 1 2 = 2 ways to select 1 penny and 1 dime. 9.
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key21fa02 - COT3100.01, Fall 2002 S. Lang (10/31/2002)...

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